Our goal in this section is to present a total ordering upon isomorphism
classes of elliptic curves in quadratic fields given a choice of integral
basis. Our method of attack will be to continually refine a partial ordering
until we have a total ordering.

Let $K$ be an abelian field, $G=\Gal(K/\QQ)$, and $\cO_K$ be our ring of
integers. We start by placing a partial ordering upon the elements of $K$.
$\CC$. If $e_k$ is the $k$-th elementary symmetric function, and $|G\alpha|$
is a vector whose components are the absolute value of elements of the orbit
of $\alpha$ under $G$ for some embedding in $\CC$, then we say that $\alpha <
\beta$ if the exists a $k$ such that $e_k(|G\alpha|) < e_k(|G\beta|)$ and
$e_i(|G\alpha|) = e_i(|G\beta|)$ for all $i > k$.\footnote{It should be noted
that the choice of embedding does not affect the ordering, since the values are
permuted by $G$.} 

It is clear that the set of elements that are incomparable with $\alpha$
include $\zeta \sigma\alpha$ for all $\zeta\in\mu_K$ and $\sigma\in G$. In
fact, it can be seen that all incomparable elements are of this form, since if
two elements are incomparable, then $|G\alpha|$ and $|G\beta|$ on all symmetric
functions on $[K:\QQ]$ variables, hence they must be equal. This implies that
$|\beta|=|\sigma\alpha|$ for some $\sigma\in G$, so since $K$ is abelian, we
know there is some $\zeta\in \mu_K$ such that $\beta = \zeta\sigma\alpha$.

This partial ordering naturally gives a partial ordering upon ideals of $O_K$,
by defining $e_k(I) = \min\{e_k(|G\alpha|) : \alpha \in I\}$ and saying that
$I < J$ if there exists a $k$ such that $e_k(I) < e_k(J)$ and $e_i(I) = e_i(J)$
for all $i > k$. In this case we can see that two ideals are incomparable if
and only if one can be obtain by applying a field automorphism to the elements
of the other.

To order between Galois conjugate ideals, we specialize to quadratic fields,
and assume that we have chosen $\{1,\gamma\}$ as our integral basis.
For each ideal $I$ we define $\psi_I:O_K/I \to \ZZ/N(R_I)\ZZ$ to be unique
homomorphism that factors through $O_K/R_I$, where $R_I = I/(I\cap \ZZ)O_K$.
We then refine our previous ordering by saying that $I < J$ if 
$\mods(\psi_I(\gamma)) < \mods(\psi_J(\gamma))$, where $\mods:\ZZ/n\ZZ\to \ZZ$
is the lift that lies in the interval $(-n/2,n/2]$. With this we have a total
ordering upon ideals of $O_K$.


